24,244 research outputs found
Exact Partition Functions for Gauge Theories on
The noncommutative space , a deformation of
, supports a -parameter family of gauge theory models with
gauge-invariant harmonic term, stable vacuum and which are perturbatively
finite to all orders. Properties of this family are discussed. The partition
function factorizes as an infinite product of reduced partition functions, each
one corresponding to the reduced gauge theory on one of the fuzzy spheres
entering the decomposition of . For a particular
sub-family of gauge theories, each reduced partition function is exactly
expressible as a ratio of determinants. A relation with integrable 2-D Toda
lattice hierarchy is indicated.Comment: 20 pages. Title modified. Typos corrected. Version to appear in
Nucl.Phys.
Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
Recent vigorous investigations of topological order have not only discovered
new topological states of matter but also shed new light to "already known"
topological states. One established example with topological order is the
valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are
disordered spin liquids with no spontaneous symmetry breaking but most
typically manifest topological order known as hidden string order on 1D chain.
Interestingly, the VBS models are based on mathematics analogous to fuzzy
geometry. We review applications of the mathematics of fuzzy super-geometry in
the construction of supersymmetric versions of VBS (SVBS) states, and give a
pedagogical introduction of SVBS models and their properties [arXiv:0809.4885,
1105.3529, 1210.0299]. As concrete examples, we present detail analysis of
supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and
UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and
four-superspheres. The SVBS states are physically interpreted as hole-doped VBS
states with superconducting property that interpolate various VBS states
depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS
states are explicitly constructed, and their gapped excitations are derived
within the single-mode approximation on 1D SVBS chains. Prominent features of
the SVBS chains are discussed in detail, such as a generalized string order
parameter and entanglement spectra. It is realized that the entanglement
spectra are at least doubly degenerate regardless of the parity of bulk
(super)spins. Stability of topological phase with supersymmetry is discussed
with emphasis on its relation to particular edge (super)spin states.Comment: Review article, 1+104 pages, 37 figures, published versio
Neutrality and Many-Valued Logics
In this book, we consider various many-valued logics: standard, linear,
hyperbolic, parabolic, non-Archimedean, p-adic, interval, neutrosophic, etc. We
survey also results which show the tree different proof-theoretic frameworks
for many-valued logics, e.g. frameworks of the following deductive calculi:
Hilbert's style, sequent, and hypersequent. We present a general way that
allows to construct systematically analytic calculi for a large family of
non-Archimedean many-valued logics: hyperrational-valued, hyperreal-valued, and
p-adic valued logics characterized by a special format of semantics with an
appropriate rejection of Archimedes' axiom. These logics are built as different
extensions of standard many-valued logics (namely, Lukasiewicz's, Goedel's,
Product, and Post's logics). The informal sense of Archimedes' axiom is that
anything can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and it
is not well-founded and well-ordered. On the base of non-Archimedean valued
logics, we construct non-Archimedean valued interval neutrosophic logic INL by
which we can describe neutrality phenomena.Comment: 119 page
The index of the overlap Dirac operator on a discretized 2d non-commutative torus
The index, which is given in terms of the number of zero modes of the Dirac
operator with definite chirality, plays a central role in various topological
aspects of gauge theories. We investigate its properties in non-commutative
geometry. As a simple example, we consider the U(1) gauge theory on a
discretized 2d non-commutative torus, in which general classical solutions are
known. For such backgrounds we calculate the index of the overlap Dirac
operator satisfying the Ginsparg-Wilson relation. When the action is small, the
topological charge defined by a naive discretization takes approximately
integer values, and it agrees with the index as suggested by the index theorem.
Under the same condition, the value of the index turns out to be a multiple of
N, the size of the 2d lattice. By interpolating the classical solutions, we
construct explicit configurations, for which the index is of order 1, but the
action becomes of order N. Our results suggest that the probability of
obtaining a non-zero index vanishes in the continuum limit, unlike the
corresponding results in the commutative space.Comment: 22 pages, 8 figures, LaTeX, JHEP3.cls. v3:figures 1 and 2 improved
(all the solutions included),version published in JHE
- …